Remote Sensing of Water and Environment

Chapter 6: Microwave Dielectric Properties of Natural Earth Materials

Ardeshir Ebtehaj
University of Minnesota
Table of Contents
Chapter 6: Microwave Dielectric Properties of Natural Earth Materials 1 Introduction 2 Pure-Water Single-Debye Dielectric Model (GHz) 3 Saline-Water Double-Debye Dielectric Model (D3M, GHz) 4. Dielectric Constant of Pure Ice 5 Dielectric Mixing Models for Heterogeneous Materials 5-1 Randomly Oriented Ellipsoidal Inclusions 5.2 Tinga-Voss-Blossey (TVB) Formulas 6 Dilectric constant of sea ice 7 Dilectric Constant of Snow 7-1 Dry snow Permittivity of dry snow Loss factor of dry snow 7-2 Wet snow 8 Dielectric Constant of Soil 8-1 Dry Soil 8-2 Wet Soil 9 Dielectric Constant of Vegetation 9-1 Dielectric Constant of Canopy Constituents 9-2 Dielectric Model Free Water Bound Water Empirical Fits

1 Introduction

In this chapter we will learn about models of ϵ for different natural materials. Natural materials can be classified into three catagories:
  1. Homogeneous substances (pure water and ice)
  2. Electrolytic solution (salt or suger in water)
  3. Heterogeneous materials (sea ice, snow, soil, vegetation)
For a medium consisting of a host material with , which is contained a homogeneous random concentration of ellipsoidal particles called inclusions with dielectric constant .
The dielectric constanct of the mixture is
where is th effective or spatial average value of the dielectric constant of the medium and is the fluctuating component.
In this chapter, we focus on attenuation as a result of and not scattering.
For a plane wave in a lossy medium, we have
where and
[Np/m]
[rad/m]
Ignoring scattering loss in the medium, one can have the following reprsentation of the power density throughout the depth of the medium
where is the power absorption coefficient
.
A related quantity of interest is penetration depth . For a scatter-free medium:
[m]
Note that , where is the skin depth.
-----------------------------------------------------
Example 5-1: For a wet snow with density , as the snow wetness chnages from 1 to 10%, the snow dielectric constant at GHz chnages from to . Compute the penetration depth and explain the difference.
Solution:
f=10; % [GHz]
omega = 2*pi*f*1e9;
c = 3e8;
k0 = omega/c;
eps_1 = 1.7-1j*0.035; % 1% wetness
eps_10 = 2.8-1j*0.92; % 10% wetness
% 1% wetness
npp_1 =-imag(sqrt(eps_1));
alpha_1 = k0*npp_1;
delta_p_1 = 1/(2*alpha_1)
 
% 10% wetness
npp_10 =-imag(sqrt(eps_10));
alpha_10 = k0*npp_10;
delta_p_10 = 1/(2*alpha_10)
% When snow becomes wet, penetration depth decreases.
-----------------------------------------------------

2 Pure-Water Single-Debye Dielectric Model (GHz)

For pure water the signle-Debye dielectric model (SD2M) is as follows:
: dielectric constant @
: dielectric constant @
: relaxation time constant [s]
f: frequency [Hz]
where
where T is in []
The term is often called relaxation frequency that is
GHz
GHz
Taking derivative of with respect to f shows that is maximum at .
1.PNG 2.PNG
Microwave spectra of the relative permittivity and loss factor of pure water at two different temperatures.

3 Saline-Water Double-Debye Dielectric Model (D3M, GHz)

The unit psu is an acronym for practical salinity unit that is equivalent to grams of salt to kg of water. The averaged salinity in the global ocean is 35.5 psu, varying from less than 15 psu at the mouth of the rivers to more than 40 psu in the Dead Sea.
We have two relaxation terms . Therefore, shows two extrems at:
The parameter functions are:
where:
The value of coefficients of D3M equation (Double-Debye dielectric model) is as follows.
1.PNG
1.PNG 2.PNG
Microwave spectra of (left) the permittivities and (right) loss factors of pure water () and seawater () at C. Sea salinity is 32.54 psu.
% D3M model
T = 20; % temperature in C
f = 5; % frequency in GHz
S = 35; % salinity in parts per thousand
 
sig_35 = 2.903602 + 8.607e-2*T + 4.738817e-4*T^2 + -2.991e-6*T^3 + 4.3041e-9*T^4;
 
P = S*((37.5109 + 5.45216*S + 0.014409*S^2)/(1004.75 + 182.283*S + S^2));
 
alpha0 = (6.9431 + 3.2841*S + -0.099486*S^2) / (84.85 + 69.024*S + S^2);
 
alpha1 = 49.843 + -0.2276*S + 0.00198*S^2;
 
Q = 1 + ((alpha0*(T-15))/(T+alpha1));
 
sigma = sig_35*P*Q;
 
 
%Other Model Paramaters
a=[0.46606917e-2 -0.26087876e-4 -0.63926782e-5 0.63000075e1 0.26242021e-2 -0.42984155e-2 ...
0.34414691e-4 0.17667420e-3 -0.20491560e-6 0.58366888e3 0.12634992e3 0.69227972e-4 ...
0.38957681e-6 0.30742330e3 0.12634992e3 0.37245044e1 0.92609781e-2 -0.26093754e-1];
 
eps_w0 = 87.85306*exp(-0.00456992*T - a(1)*S - a(2)*S^2 - a(3)*S*T);
eps_w1 = a(4)*exp(-a(5)*T-a(6)*S-a(7)*S*T);
tau1 = (a(8)+a(9)*S)*exp(a(10)/(T+a(11)));
tau2 = (a(12)+a(13)*S)*exp(a(14)/(T+a(15)));
epsInf = a(16) + a(17)*T + a(18)*S;
 
%Complex Permitivity Calculation
eps = epsInf + ((eps_w0-eps_w1)./(1-1j*2*pi.*f.*tau1)) + ((eps_w1-epsInf)./(1-1j*2*pi.*f.*tau2)) + 1j*((17.9751*sigma)./f);
 
 
epsr = real(eps) % real part of relative dielectric constant
epsr = 65.7673
epsi = imag(eps) % imaginary part of relative dielectric constant
epsi = 34.2531

4. Dielectric Constant of Pure Ice

Unlike liquid water, whose relaxation frequencies lie in the microwave region, the relaxation frequency of pure ice occurs in the kilohertz region.
Thus, the pervious equation in MW band simplify as follows:
since the single the Debye model expressions simplifies to:
where .
It was shown that is independent of frequency in MW bands and the following model represents its dependence on the temperature.
where T is in . Practically, we can ignore the dependancy to temperature and use across the entire MW spectrum.
There are more accurate model for that accounts for infrared absorption spectrum as follows.
where:
f is in GHZ.
1.PNG
Loss factor of pure ice.
Note: Fresh water ice often contains ionic impurities, such as dissolved salts, which causes its to increase significantly in comparison with that for pure ice. Observational evidence suggests that can be 2-8 times longer than that of pure ice.
%% Relative Dielectric Constant of Pure Ice
 
% frequency 1<f<1000GHz
 
% Input Variables:
% T: Temperature in degree C
% f: frequency in GHz
 
% Output Variables:
% epsr: real part of relative dielectric constant
% epsi: imaginary part of relative dielectric constant
%------------------------------------------------------------
T = -1; % [C]
f = 100; % [GHz]
 
T = T + 273; % represent temperature in Kelvin
 
theta = (300/T) - 1;
 
B1 = 0.0207;
B2 = 1.16e-11;
b = 335;
 
alpha = (0.00504 + 0.0062*theta)*exp(-22.1*theta);
 
betaM = (B1/T) * exp(b/T) / (exp(b/T)-1)^2 + B2.*f.^2;
delBeta = exp(-9.963 + 0.0372*(T-273.16));
 
beta = betaM + delBeta;
 
% epsr = 3.1884 + 9.1e-4*(T-273) .*f./f;
 
epsr = 3.1884 + 9.1e-4 *(T-273)
epsr = 3.1875
 
epsi = alpha./f + beta.*f
epsi = 0.0090

5 Dielectric Mixing Models for Heterogeneous Materials

A mixed material is defined by the host and inclusion materials.
Dilectric constant of a mixture depends on:
We need to find an average representation of dilectric constant based on the dilectric values of the host and inclusions.
Note: Here we assume that the dimensions of the inclusions are much smaller than the wavelength of the EM propagation field, thereby allowing to ignore volume scattering.

5-1 Randomly Oriented Ellipsoidal Inclusions

A host material containing only one type of inclusion is called a two-phased mixture consider that inclusions are ellipsoidal particles randomly dispersed within the host material. Each ellipsoid has dimension 2a, 2b, 2c along its major axes and host and inclusion material, have isotropic dilectric constants , respectively. The inclusion concentration in a unit volume is defined by inclusion volume fraction as follows:
Thus fraction of the host material is , where N is the number of ellipsoids per the volume.
1.PNG
Randomly oriented ellipsoidal inclusions of identical size, shape, with dielectric constant , dispersed in a host material with dielectric constant . The ellipsoidal particles have semiaxes a, b, and c.

5.2 Tinga-Voss-Blossey (TVB) Formulas

The general dilectric formulation pertains to randomly dispersed confocal ellipsoids consisting of inner ellipsoid with dilectric constant as shown in the following figure.
1.PNG
Two-phase confocal ellipsoidal TVBmodel (Tinga et al., 1973) in which each ellipsoidal inclusion is surrounded by a shell of host material whose thickness is governed by the volume fraction of the host material.
Note: As is evident in all cases as then and when then
It is important to note that there are numerous models of dilectric constant of a mixture and all of them are in the following form:
where
linear model
refractive model
cubic model.

6 Dilectric constant of sea ice

Little progress has been made in chracterization of the dielectirc constant of sea ice due to its complexity.
Hetrogeneity of the inclusion varies in space and time significantly as the ice evolves
The first two categories are similar in tems of the ice structure and dielectric properties. However, the brine concentration is different between multi-year sea ice and other two catagories, which is one of the main differences between them in terms their dilectric constant.
In general is a function of the following parameters:
It is important to know that is a function of tempreture. The reason is that because the temprerature drop causes more pockets of brine inclusions to freeze up and so the volume fraction of brine changes.
Temperature variation of the permittivity Temperature variation of the loss factor
for three types of sea ice at 10 GHz. of three types of sea ice at 10 GHz.
Measured permittivity and loss factor of simulated sea ice at 4.7 GHz, plotted as a function of brine volume fraction
Summary from the available observational data:

7 Dilectric Constant of Snow

We treat dry and wet snow separatly. The reason is that permittivity of ice () and water () are drasically different. Thus a small amount of liquid water can change the dielectric constant of snow significantly.

7-1 Dry snow

Permittivity of dry snow

The dielectric constant of dry snow depends on:
,
Assuring (air) and grains of snow are circular, model results in:
For pure ice, we have and thus we have,
,
which is independent of temperature and frequency and only depends on the density.
1.PNG
Measured permittivity of dry snow as a function of snow density over frequency range 0.8--37 GHz.
An equally good fit to the data is provided (Matzler, 2006) as follows:

Loss factor of dry snow

When, ice is not pure, the loss factor is non-zero and depends on both temperature and frequency. Using TBV models and assuming:
Considering , we have:
Note that dry snow is a low-loss medium if . Dry snow becomes more lossy as its density increases.
1.PNG
% The code computes the real and imaginary parts of the relative dielectric constant of Dry Snow
 
% Input Variables:
% T: Temperature in C
% Ps: Dry Snow Density in g/cm^3
% f: frequency in GHz (0.8 --37 GHz)
% Output Products:
%epsr: real part of relative dielectric constant
%epsi: imaginary part of relative dielectric constant
rho_s = 0.5;
T = -10;
f = 6; % [GHz]
vi= rho_s/0.9167;
 
[epsr_ice, epsi_ice] = RelDielConst_PureIce(T,f);
 
if vi <= 0.45
epsr_ds = 1 + 1.4667 .* vi + 1.435 .* vi.^3; % 0<vi<0.45
else
epsr_ds = (1+ 0.4759 .*vi).^3; % vi>0.45
end
 
epsi_ds = 0.34 * vi * epsi_ice ./(1- 0.42 * vi).^2;
epsr_ds
epsr_ds = 1.9983
epsi_ds
epsi_ds = 1.5365e-04

7-2 Wet snow

In proximity of , snow can hold water in liquid form , where is liquid water content of snow (volume fraction of water in snow).
An extensive investigation of the dielectric behavior of natural snow was conducted by Hallikainen et al. (1983, 1984, 1986) using a free-space transmission technique. The permittivity and the dielectric loss factors of wet snow, and , were measured for 110 snow samples at nine frequencies between 4 and 18 GHz. Of these, 62 samples were also measured at 3 GHz and 37 GHz. Thus, although their results cover the range between 3 and 37 GHz, they are heavily influenced by the 4-18 GHz measurements. The range of covered by the samples extended between 1 and 12 percent.
Based on these measurments, a Debye-like model has been proposed for computation of wet snow dielectric constant as follows.
1.PNG 2.PNG
Real and imaginary parts of the relative dielectric constant of snow as a function of liquid-water content at 6 GHz.
3.PNG
Spectral variation of the permittivity of wet snow with snow wetness as a parameter. The plots are based on the modified Debye-like model.
%%% Dilectric constant of wet snow.
 
% Input Variables:
% rho_s: Snow Density (g/cm^3)
% mv: volumetric water content (0<mv<30 in percentage scale)
% f: frequency in GHz
% Output Products:
% epsr: real part of relative permitivity
% epsi: imaginary part of relative permitivity
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
rho_s = 0.4;
mv = 10;
f = 1;
 
A1 = 0.78 + 0.03 .*f - 0.58e-3 .* f.^2;
A2 = 0.97 - 0.39e-2 .*f + 0.39e-3 .* f.^2;
B1 = 0.31 - 0.05 .*f + 0.87e-3 .* f.^2;
 
A = A1 .*(1.0 + 1.83*rho_s + 0.02*mv^1.015) + B1;
B = 0.073 .*A1;
C = 0.073 .*A2;
x = 1.31;
f0 = 9.07;
 
epsr = A + (B .* mv^x) ./ (1+(f/f0).^2)
epsr = 3.0223
epsi = (C .* (f/f0) .* mv^x) ./ (1+(f/f0).^2)
epsi = 0.1569
 

8 Dielectric Constant of Soil

8-1 Dry Soil

In the absence of liquid water, the microwave dielectric constant of soil, , is essentially independent of both temperature and frequency. Moreover, and . Based on experimental measurements of several soil types, Dobson et al. (1985) determined that can be modeled as
where is the soil bulk density, where is the mass of the dry soil and is the total volume.

8-2 Wet Soil

A wet-soil medium is a mixture of soil particles, air pockets, and liquid water.
The water contained in the soil usually is divided into two fractions: (a) bound water, and (b) free water.
Bound water refers to water molecules that are contained in the first few molecular layers surrounding the soil particles and therefore are tightly held by the soil particles due to the influence of matric and osmotic forces. Because the matric forces acting on a water molecule decrease rapidly with increasing distance from the soil particle, water molecules that are located several molecular layers away from soil particles are able to move within the soil medium with relative ease, and hence are referred to as “free” water.
The amount of water contained in the molecular layer adjoining the soil particles is directly proportional to the total surface area of the soil particles contained in a unit volume. The total surface area of the particles is, in turn, a function of the size distribution and mineralogy of the soil particles. For the most part, a soil is assigned to a textural class on the basis of its particle-size distribution.
Soil particles are classified as sand, silt, or clay, according to thier size. Because all soils contain a distribution of particle sizes, it is convenient to classify a soil by the weight-percent of the soil within each specific size category (sand, silt, or clay).
The two terms commonly used to characterize the moisture content of a soil sample are volumetric moisture and gravimetric moisture ,
Dielectric constant of wet soil dpends on
Measured dielectric constant for five soils at 5 GHz (left) and measured dielectric constant as a function of volumetric moisture content for a loamy soil at four microwave frequencies (right) [Hallikainen et al., 1985].
Measured (left) permittivity and (right) loss factor of loamy soil as a function of frequency with volumetric moisture content as a parameter [Hallikainenet al., 1985].
Measured permittivity and loss factor of silt loam soil as a function of frequency with temperature as a parameter [from Hallikainen et al., 1984b].
Dobson et al. (1985) used a four-component model consisting of solid soil material, air, bound water, and free water.
where is given by single Deby model as follows
,
The expressions for , and are given previously and and
where and are mass fraction of sand and clay.
Peplinski et al. (1995) conducted a study of several soil types, with specific focus on the dielectric behavior in the 0.3--1.3 GHz range abd suggested the following formula for the conductivity parameter:
%% dielectric constant of soil at a given temperature 0<T<40C,
% frequency, volumetric moisture content, soil bulk density, sand and clay fractions.
 
% Input Variables:
% f: frequency in GHz
% T: temperature in C
% rho_b: bulk density in g/cm3 (typical value is 1.7 g/cm3)
% S: Sand Fraction ( 0<S< 1)
% C: Clay Fraction ( 0<C< 1)
% mv: Volumetric Water content 0<mv<1
 
% Outputs:
%epsr: real part of dielectric constant
%epsi: imaginary part of dielectric constant
clear
f=1.4;
mv = 0.30;
rho_b = 1.7;
f_hz = f * 1.0e9; % transform from GHz to Hz
S = 0.3;
C = 0.5;
T = 25;
%%
beta1 = 1.27 - 0.519 * S - 0.152* C;
beta2 = 2.06 - 0.928 * S - 0.255 * C;
alpha = 0.65;
 
eps_0 = 8.854e-12;
 
sigma_s = 0;
 
if f > 1.3
sigma_s = -1.645 + 1.939 * rho_b - 2.256*S + 1.594 * C; % Dobson et al. (1985)
end
if f >= 0.3 && f <= 1.3
sigma_s = 0.0467 + 0.22 * rho_b - 0.411*S + 0.661 *C; % Peplinski et al. (1995)
 
end
 
%Dielectric Constant of Pure Water
ew_inf = 4.9;
ew_0 = 88.045 - 0.4147 * T + 6.295e-4 * T^2 + 1.075e-5 * T^3;
tau_w = (1.1109e-10 - 3.824e-12*T +6.938e-14*T^2 - 5.096e-16*T^3)/2/pi;
 
epsrW = ew_inf +(ew_0-ew_inf)./(1 + (2*pi*f_hz*tau_w).^2);
 
epsiW = 2*pi*tau_w .*f_hz *(ew_0-ew_inf) ./(1 + (2*pi*f_hz*tau_w).^2) + ...
(2.65-rho_b)/2.65/mv * sigma_s ./(2*pi*eps_0*f_hz);
% calculating dielectric constant of soil
epsr = (1+ 0.66*rho_b + mv^beta1 * epsrW.^alpha - mv).^(1/alpha)
epsr = 18.5743
epsi = mv^beta2 .* epsiW
epsi = 4.4194
Comparison of values of and measured at 0.3 and 1.3 GHz with calculations based on Model 1 by Dobson (1985) and Model 2 by Peplinski et al. (1995).

9 Dielectric Constant of Vegetation

From the standpoint of wave propagation, a vegetation canopy is a dielectric mixture consisting of discrete dielectric inclusions (such as leaves, stalks, and fruit) distributed in a host material (air). In most canopies, the sizes of the inclusions are either comparable to or larger than the wavelength in the microwave region, which means that the canopy is an inhomogeneous, anisotropic medium. Propagation through such a medium entails both absorption and scattering.

9-1 Dielectric Constant of Canopy Constituents

Dielectric measurements of several grain types have been reported over a wide range of frequency band to 12.2 GHz in the microwave region (Nelson and Stetson, 1976).
Measured dielectric constant of red winter wheat heads as a function of moisture content [from Nelson and Stetson, 1976].
The moisture content of a vegetation material, such as leaves or stalks, is measured on a gravimetric wet weight basis () or on a volume basis (). The two quantities are related by
where is the dry density of the solid material. A typical value of for leaves is 0.3 . The range of extends from 0 to about 0.9, and the corresponding range of is from 0 to 0.7 (for ≈ 0.3).
Using a waveguide transmission technique, Ulaby and Jedlicka (1984) measured the dielectric properties of leaves and stalks of corn and wheat over the 1.1 to 8.4 GHz range. Among the important parameters governing the behavior of and is the salinity S of the fluid samples extracted from the vegetation material.
Measured moisture dependence of the dielectric constant of corn leaves at 1.5, 5.0, 8.0 GHz [Ulaby and Jedlicka, 1984].
Family of dielectric spectra for corn leaves at T = 22 ◦C and various moisture contents [El-Rayes and Ulaby, 1987].
Variation of er of corn leaves with decreasing temperature from 22 down to −32 C [El-Rayes and Ulaby, 1987].

9-2 Dielectric Model

Ulaby and El-Rayes (1987) introduced a linear model for the dielectric constant of vegetation in the form of an additive mixture of three components:
where is a nondispersive residual component to be determined empirically, and are the complex dielectric constants of free water and bound water, respectively, and and are their associated volume fractions.

Free Water

For C, the relaxation frequency of water is 18 GHz and thus

Bound Water

Empirical Fits

By fitting the measured dielectric data to the dielectric model, the remaining quantities were found to have the following form
% Code computes the real and imaginary parts of the relative dielectric constant of vegetation material, such as corn leaves, in the microwave region.
% Input Variables:
%f: frequency in GHz
%mg: Gravimetric moisture content 0< mg< 1
% Output Products:
%eps_v_r: real part of dielectric constant
%eps_v_i: imaginary part of dielectric constant
S = 1; % salinity [psu]
mg = 0.4;
f = 1.4;
%-- free water in leaves
 
sigma_i = 0.17.*S - 0.0013 .* S.^2;
 
eps_w_r = 4.9 + 74.4 ./( 1 + (f/18).^2);
 
eps_w_i = 74.4 .*(f/18) ./( 1 + (f/18).^2) + 18*sigma_i ./f ;
 
% bound water in leaves
 
eps_b_r = 2.9 + 55*(1+ sqrt(f/0.36))./( (1+ sqrt(f/0.36)).^2 + (f/0.36));
 
eps_b_i = 55*sqrt(f/0.36) ./ ( (1+ sqrt(f/0.36)).^2 + (f/0.36));
 
 
% empirical fits
 
v_fw = mg .*( 0.55 * mg - 0.076);
v_bw = 4.64 .*mg.^2 ./(1 + 7.36 * mg.^2);
 
eps_r = 1.7 - 0.74 *mg + 6.16 .* mg.^2;
 
 
eps_v_r = eps_r + v_fw .* eps_w_r + v_bw .*eps_b_r
eps_v_r = 12.3007
eps_v_i = v_fw .* eps_w_i + v_bw .*eps_b_i
eps_v_i = 3.3628